Wave phenomena in radar meteorology Chris Westbrook

Reflectivity, Z - Measure of intensity reflected back to radar: back scatter cross section (intensity) radar wavelength Doppler radar Measure frequency shift (the police car effect) – gives reflectivity weighted average fall speed: Includes (unknown) contribution from velocity of the air (updraft/downdraft) - can remove this effect using two wavelengths (soon!) Rayleigh (small size compared to wavelength) Almost no phase difference (uniform E - field) across particle: wave blind to details, scatters Isotropically according to how much material is there eg. Rain Radar ~10cm drop~1mm RAL Chilbolton

Doppler velocity RED = TOWARDS RADAR BLUE = AWAY FROM RADAR Reflectivity RED / PURPLE = HEAVY RAIN Strong ascending motion can be seen in the regions of heaviest precipitation. At the tropopause, the cloud spreads out horizontally to form cirrus anvil clouds. EXAMPLE: THUNDERSTORM 28 th JULY 2000

phase shift Z DR 0 dB (Z H = Z V ) 1 mm 3 mm 4.5 mm Z DR = 1.5 dB (Z H > Z V ) Z DR = 3 dB (Z H >> Z V ) Differential polarisability Bigger drops arent spherical, but oblate (pancake) drop is more easily polarised in horizontal direction than in the vertical, so Z H > Z V Look at ratio Z DR provides estimate of drop size Differential phase shift dp Flattened drops - Horizontally polarised component is slowed down more than the vertically polarised component difference in phase between H and V Polarisation measurements Less influenced by largest drops, but data can be noisy. Helps distinguish big rain drops from hailstones (which are spherical and so dp =0)

ICE RAIN ICE RAIN Observations Cold front 20 th October 2000 (time scale ~ 1¼ hours) melting layer

Clear air returns Refractive index variations in clear air can produce radar returns if length scale is /2 Waves from each layer add up constructively and in phase (like Bragg scattering in crystals) Allows you to see the boundary layer, Edges of clouds Turbulence. insects? Boundary layer Bulge, indicative of storm

Ice clouds Want to interpret observations in terms of : how much ice is in the cloud, how big the ice particles are, how fast theyre falling etc… Cover about ¼ of earths surface typically – important for radiation budget / climate etc. Need to model the scattering properties of ice particles in clouds

Pristine crystals – Columns, Plates, Bullet rosettes First few hundred metres AGGREGATES - (complicated!) Lower Altitudes Diffusion of water vapour onto ice Sedimentation at different speeds Cloud radars: wavelength and ice particle size are comparable PARTICLE SHAPE MATTERS! Previous studies concentrate only on pristine crystals We want to try and model the scattering from aggregates Timely, since next month CloudSat 3mm radar will be in space.

Aggregation model relative fall speedtotal possible collision area Mean field approach – big box of snowflakes, pick pairs to collide with probability proportional to: Then, to get the statistics right, pick a random trajectory from possible ones encompassed by and track particles to see if they actually do collide – if so, stick them together. UNIVERSALITY: Statistically self-similar structure – fractal dimension of 2 Also self-similar size distribution. Real ice aggregates from a cirrus cloud In the USA Simulated aggregates (aggregates of bullet rosette crystals)

Rayleigh-Gans theory (Born approximation) r dvdv k Assume: 1. each volume element sees only the applied wave 2. the elements scatter in the same way as an equivalent volume sphere (amplitude ~ dv / 2 ) NO INTERACTION BETWEEN VOLUME ELEMENTS. So just add up the scattered amplitude of each element × a phase factor exp(I 2kr) … [ Small particle limit (kr 0) reduces to Rayleigh sphere formula, ie. intensity ~ volume 2 / 4 ] For small particles, wave only sees particle volume, not particle shape. (Rayleigh regime) Phase shift between centre of mass and element at position r is kr So for back-scatter the total phase difference in the scattered wave is 2kr (ie. there and back) If particle size and wavelength are comparable (cloud radar / ice particles) then we need a more sophisticated theory… |k| = APPROXIMATE PARTICLE BY ASSEMBLY OF SMALL VOLUME ELEMENTS dv

So the scattered intensity (radar cross section) is: Dimensionless function f - tells you the deviation from the Rayleigh formula with Increasing size r The form factor f is easy to calculate, and allows you (for a given shape) to parameterise the scattering in terms of: 1.the particle volume, 2.characteristic particle length r (relative to the wavelength). f = / (Rayleigh formula) to appear in the January edition of Q. J. Royal Met. Soc. (Westbrook CD, Ball RC & Field PR Radar scattering by aggregate snowflakes) LIMITATION OF RAYLEIGH-GANS: Assumes no interaction between volume elements (low density / weak dielectric / small kr limit) Good for first approximation, but is this really all ok? Universal form factor for aggregates irrespective of the pristine crystals that compose them (as long as crystals much smaller than wavelength)

The discrete (coupled) dipole approximation DDA applied Each dipole is polarised in response to: 1.The incident applied field 2.The field from all the other dipoles Approximate particle by an assembly of polarisable, INTERACTING dipoles: etc.. Now instead of simple volume integral of Rayleigh-Gans, have 3N coupled linear equations to solve: polarisability of dipole k Electric field at j Applied field at jTensor characterising fall off of the E field from dipole k, as measured at j Electric field at k

Aggregates of: 100 m hexagonal columns, aspect ratio = 1/2 discrete dipole approximation (Increased backscatter relative to Rayleigh-Gans) Rayleigh-Gans f = Ratio of real backscatter to Rayleigh formula w l w/l =½ Results Would like to parameterise the increased backscatter – must depend on: 1.Volume fraction of ice ie. Volume / (4 r 3 /3) 2.Size of aggregate relative to wavelength kr USE DDA CALCULATIONS TO WORK OUT FUNCTIONS S AND -THEN ONLY NEED r AND v TO CACLULATE THE SCATTERING.

Acknowledgements / Ads etc. John Nicol for the clear air boundary layer Robin Hogan for advice and an old presentation from which the animations were Robin Ball (Physics, Warwick) and Paul Field (NCAR), collaboration on ice aggregation & Rayleigh-Gans work. Photographs of pristine snow crystals were from (Caltech) For more information on the radar group at Reading: and on my work… Chris Westbrook Room 2U04, Meteorology